elmore_straight_wire.m

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% Elmore delay sizing for a straight wire.
% (a figure of area-delay tradeoff is generated)
%
% This is an example taken from the EE364 lecture notes:
%
%   "Problems in VLSI design" lecture by Prof. Boyd
%   Available at: http://www.stanford.edu/class/ee364
%
% We consider the problem of finding optimal width profile
% for a straight wire segmented into N parts. We want to
% minimize the Elmore delay, subject to limits on wire width
% and the total area. We use a pi-model for each wire segment.
% Problem can be formulated as GP:
%
%   minimize   D
%       s.t.   w_min <= w <= w_max
%              area  <= Amax 
%
% where variables are widths w (and arrival times T that are used
% to formulate the overall delay D expression).
%
% Important: We label root node as 1, and all the other nodes as
%            node_label_in_the_paper + 1 (due to Matlab's convention).
%            Also label nodes with increasing numbers downstream.
%
% Almir Mutapcic 02/01/2006
clear all; close all;
PLOT_TRADEOFF = 1; % to disable set PLOT_TRADEOFF = 0;

%********************************************************************
% user supplied data (problem constants and tree topology)
%********************************************************************
N = 10+1; % number of segment (including the root node which is labeled as 1)

% parent node array for the straight wire
% specifies which node is a unique parent for node i (always have a tree)
parent = [0:N-1]; 

% problem constants
Rsource = 0.1;
l = 1*ones(N-1,1);
alpha = 1*ones(N-1,1);
beta  = 1*ones(N-1,1);
gamma = 1*ones(N-1,1);

% load capacitance at each node
Cload = [0; ones(N-1,1)]; 

% minimum and maximum width and area specification 
Wmin = 1;
Wmax = 10;
Amax = 50;

%********************************************************************
% derived data (computed from user's data)
%********************************************************************
% compute children cell array (evaluate who are children for each node)
children = cell(N,1);
leafs = [];
for node = [1:N]
  children{node} = find(parent == node);
  if isempty(children{node})
    leafs(end+1) = node; % leafs have no children
  end
end

%********************************************************************
% optimization
%********************************************************************
% optimization variables
gpvar w(N-1)     % wire width
gpvar T(N)       % arrival time (Elmore delay to node i)

% wire segment resistance is inversely proportional to widths
R = alpha.*l./w;
R = [Rsource; R];

% wire segment capacitance is an affine function of widths
C_bar = beta.*l.*w + gamma.*l;
C_bar = [0; C_bar];

% compute common capacitances for each node (C_tilde in GP tutorial)
C_tilde = posynomial; % initialize an empty posynomial
for node = [1:N]
  C_tilde(node,1) = Cload(node);
  for k = parent(node)
    if k > 0; C_tilde(node,1) = C_tilde(node,1) + C_bar(k); end;
  end
  for k = children{node}
    C_tilde(node,1) = C_tilde(node,1) + C_bar(k);
  end
end

% now compute total downstream capacitances
C_total = posynomial; % initialize an empty posynomial
for node = N:-1:1
  C_total(node,1) = C_tilde(node);
  for k = children{node}
    C_total(node,1) = C_total(node,1) + C_total(k,1);
  end
end

% generate Elmore delay constraints
elm_delay_constr = [R(1)*C_total(1) <= T(1,1)];
for node = 2:N
  elm_delay_constr = [elm_delay_constr; ...
                      R(node)*C_total(node) + T(parent(node),1) <= T(node,1)];
end

% collect all the constraints
area = sum(w.*l);
constr(1) = area <= Amax;
constr = [constr; Wmin*ones(N-1,1) <= w; w <= Wmax*ones(N-1,1)];
constr = [constr; elm_delay_constr];

% objective is the critical Elmore delay
D = max( T(leafs) );

% solve the problem
[D_value, solution, status] = gpsolve(D, constr);
assign(solution);

% save for plotting
ckt_delay_plot = D_value;
Amax_plot = Amax;

fprintf(1,'\nOptimal Elmore delay for Amax = %d is %3.4f.\n', ...
        Amax, D_value)
disp('Optimal wire widths are: '), w

%********************************************************************
% tradeoff curve code
%********************************************************************
if( PLOT_TRADEOFF )

% set the quiet flag (no solver reporting)
global QUIET; QUIET = 1;

disp('generating the tradeoff curve')
Darray = []; widths = [];
for Amax = [10.01 10.05 10.5 11 12:2:20 22.5 25:5:60]
  % formulate the GP problem and solve it
  constr(1) = area <= Amax;
  [D_value, solution, status] = gpsolve(D, constr);
  Darray = [Darray D_value];
  widths = [widths solution{2,2}];
end

% enable solver reporting again
global QUIET; QUIET = 0;

% indices of four taper designs on the tradeoff curve
Amax = [10.01 10.05 10.5 11 12:2:20 22.5 25:5:60];
A11ind = find(Amax == 11);
A20ind = find(Amax == 20);
A35ind = find(Amax == 35);
A50ind = find(Amax == 50);

% plot the tradeoff curve
plot(Darray,Amax, ...
     Darray(A11ind),Amax(A11ind),'ro',...
     Darray(A20ind),Amax(A20ind),'ro',...
     Darray(A35ind),Amax(A35ind),'ro',...
     Darray(A50ind),Amax(A50ind),'ro');
xlabel('Elmore delay D'); ylabel('Amax');

% plot four taper designs
w1 = widths(:,A50ind);
w2 = widths(:,A35ind);
w3 = widths(:,A20ind);
w4 = widths(:,A11ind);
plot_four_tapers(w1,w2,w3,w4);

end
% end tradeoff curve code